Limits, Continuity & Differentiability Question - None

Question

If the function $f(x) = x^3 + e^{x/2}$ and $g(x) = f^{-1}(x)$, then the value of $g'(1)$ is:

Accepted Answer

Inverse Function Derivative — JEE Advanced 2009
Find $g(1)$: solve $f(t)=1\Rightarrow t^3+e^{t/2}=1$. Test $t=0$: $0+e^0=1$ ✓. So $g(1)=0$.
$f'(x)=3x^2+\dfrac{1}{2}e^{x/2}$. At $x=0$: $f'(0)=0+\dfrac{1}{2}=\dfrac{1}{2}$.
$g'(1)=\dfrac{1}{f'(g(1))}=\dfrac{1}{f'(0)}=\dfrac{1}{1/2}=2$.
Answer: 2
Key concept: Always find the pre-image: $t=0$ satisfies $f(0)=1$ cleanly. Then $g'(y)=1/f'(t)$.
Trap: Solving $f'(t)=1$ (derivative equation) instead of $f(t)=1$ (value equation) to find $g(1)$.

Question:

Step-by-Step Solution

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